This problem’s probably shown up before by a different name or different variation, but I thought it was interesting. It goes like this.
I have a block of wood with integer dimensions. If I saw off two units from each dimension, the volume of the woodblock reduces by half. How many possible sets of dimensions are there for the original block? Or, what integers satisfy 2xyz = (x+2)(y+2)(z+2)?
You can see a program generate them here. There’s eighteen but I could have sworn there were twenty, when I made a similar program some years ago.
The problem changes when you vary the proportion and the number of units cut off from each dimension. (If the proportion is p, 1/p is the how much of the volume is left after cutoff.)
When c=1 and as p increases, there are less and less solutions. 2,3,4 (1,2,3) is the only solution when c=1 and p=4. No solutions when c=1 and p>=5.
When p=2 and as c increases, there are more and more and larger solutions.
It appears that the trend when p=c and both increase is that there are less solutions, though since they become large, and because waiting for it to process is a pain for high maximum values, I’m not sure about this one. With max of 1000 and p=c=11, it didn’t find any solutions.
Let me know if you find anything interesting. :)